Chapter 3: Q18MP (page 186)

Find the eigenvalues and eigenvectors of the matrices in the following problems.
$[\begin{matrix} 1 & 0 \\ 3 & - 2 \end{matrix}]$

Short Answer

Expert verified

The eigenvalues of the matrix are, $λ = 1, - 2$

The eigenvector corresponding to $λ = 1$ is $[\begin{matrix} 1 \\ 1 \end{matrix}]$

The eigenvector corresponding to $λ = - 2$ is .role="math" localid="1664274394944" $[\begin{matrix} 0 \\ 1 \end{matrix}]$

Step by step solution

To find the eigenvalues of matrix

Let $[\begin{matrix} 1 & 0 \\ 3 & - 2 \end{matrix}]$ be the matrix.

The characteristic equation is given by,

$\begin{array}{rcl} |M - λ I| & = & 0 \\ |\begin{matrix} 1 - λ & 0 \\ 3 & - 2 - λ \end{matrix}| & = & 0 \\ (1 - λ) (- 2 - λ) & = & 0 \\ λ & = & 1, - 2 \end{array}$

Thus, the eigenvalues of the matrix are, $λ = 1, - 2$

To find the eigenvectors of matrix

Now, to find eigenvectors.

Consider $(M - λ I) X = 0$

For $λ = 1$

$\begin{array}{rcl} (M - I) X & = & 0 \\ [\begin{matrix} 0 & 0 \\ 3 & - 3 \end{matrix}] [\begin{matrix} x \\ y \end{matrix}] & = & 0 [\frac{R_{2}}{3}] \\ [\begin{matrix} 0 & 0 \\ - 1 \end{matrix}] [\begin{matrix} x \\ y \end{matrix}] & = & 0 \\ x - y & = & 0 \\ x & = & y \end{array}$

Therefore,

$X = [\begin{matrix} x \\ y \end{matrix}] X = x [\begin{matrix} 1 \\ 1 \end{matrix}]$

Hence, the eigenvector corresponding to $λ = 1$ is $[\begin{matrix} 1 \\ 1 \end{matrix}]$ .

For $λ = - 2$

$\begin{array}{rcl} |M + 2 I| X & = & 0 \\ [\begin{matrix} 3 & 0 \\ 3 & 0 \end{matrix}] [\begin{matrix} x \\ y \end{matrix}] & = & 0 (R_{2} - R_{1}) \\ [\begin{matrix} 3 & 0 \\ 0 & 0 \end{matrix}] [\begin{matrix} x \\ y \end{matrix}] & = & 0 (\frac{R_{1}}{3}) \\ [\begin{matrix} 1 & 0 \\ 0 & 0 \end{matrix}] [\begin{matrix} x \\ y \end{matrix}] & = & 0 \\ x & = & 0 \\ X & = & [\begin{matrix} 0 \\ y \end{matrix}] \\ X & = & y [\begin{matrix} 0 \\ 1 \end{matrix}] \end{array}$

Hence, the eigenvector corresponding to $λ = - 2$ is $[\begin{matrix} 0 \\ 1 \end{matrix}]$ .

Unlock Step-by-Step Solutions & Ace Your Exams!

Full Textbook Solutions
Get detailed explanations and key concepts
Unlimited Al creation
Al flashcards, explanations, exams and more...
Ads-free access
To over 500 millions flashcards
Money-back guarantee
We refund you if you fail your exam.

Start your free trial

Over 30 million students worldwide already upgrade their learning with Vaia!

Recommended explanations on Physics Textbooks

View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Find the eigenvalues and eigenvectors of the matrices in the following problems.
$[\begin{matrix} 1 & 0 \\ 3 & - 2 \end{matrix}]$

Short Answer

Step by step solution

To find the eigenvalues of matrix

To find the eigenvectors of matrix

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Physics Textbooks

Atoms and Radioactivity

Fundamentals of Physics

Electrostatics

Physical Quantities And Units

Energy Physics

Physics of Motion

Study anywhere. Anytime. Across all devices.

Find the eigenvalues and eigenvectors of the matrices in the following problems.[103-2]

Short Answer

Step by step solution

To find the eigenvalues of matrix

To find the eigenvectors of matrix

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Physics Textbooks

Atoms and Radioactivity

Fundamentals of Physics

Electrostatics

Physical Quantities And Units

Energy Physics

Physics of Motion

Study anywhere. Anytime. Across all devices.

Find the eigenvalues and eigenvectors of the matrices in the following problems.
$[\begin{matrix} 1 & 0 \\ 3 & - 2 \end{matrix}]$